At least one property of a fluid-suspended particle can be determined by combining one or more measured velocities of the particle with suitable theory relating particle and fluid properties to particle and fluid velocities and to one or more external forces acting on the particle. Herein, the term particle denotes a small body composed of solid or liquid or gas material or of multiple materials, such as a dust, smoke or pollen particle, a bacterial cell, a water or oil droplet, or an air bubble. The term particle shall also be understood herein to denote any larger body for which the present invention can measure two or more motion parameters for motion in a specified direction at or near a specified time, a larger body such as a raindrop, vehicle or satellite. The term fluid as used herein denotes a gas, a rarefied gas or vacuum, or a liquid material, or a mixture thereof, and also a suspension of solid or liquid or gas bodies, or bodies of some combination thereof, in a gas or liquid material. However, in each case it is required that a reflected, scattered or emitted signal from a suspended particle species of interest be discriminated from any signals from suspending fluid material. The term suspending shall denote herein the sense of containing and not necessarily the sense of supporting or holding.
Theory relating particle and suspending fluid velocities and particle and fluid properties is described by N. Fuchs in his seminal book The Mechanics of Aerosols, reprinted by Dover Publications, New York, especially in chapters 2, 3, and 4 and by S. K. Friedlander in his book entitled Smoke, Dust and Haze, published by Wiley-Interscience, New York. These books and numerous others on physics, fluid mechanics, particles, colloids and aerosols provide equations by which measured particle motions can be used to determine motions of a suspending fluid, particle or fluid properties, or external forces acting on a particle.
Current techniques for measuring particle velocity include time-of-flight velocimetry (TOFV) or anemometry (TOFA) and laser Doppler velocimetry (LDV) or anemometry (LDA). A TOFV type method is described by Barton E. Dahneke in U.S. Pat. No. 3,854,321. A TOFA method is described by L. Lading, A. Skov Jensen, C. Fog and H. Anderson in the journal article "Time-of-flight Laser Anemometry for Velocity Measurements in the Atmosphere" published in Applied Optics, volume 17, number 10, page 1486. LDV and LDA techniques are described by F. Durst, A. Melling and J. H. Whitelaw in the book Principles and Practices of Laser Doppler Anemometry, second edition, published by Academic Press in 1981. These TOFV, TOFA, LDV, and LDA techniques have proven useful in determining fluid velocity because they are non-intrusive, non-destructive, simple, rapid, and, oftentimes, accurate.
However, there are problems with the current TOFV, TOFA, LDV, LDA and similar techniques in measurement of particle and fluid velocities and properties. One such problem is the provision by current methods of only a particle velocity component in one or more specified directions, information inadequate for characterizing highly time- or location-varying particle or fluid motion. To characterize such rapidly changing motions of a particle or fluid, velocity and acceleration, and sometimes higher-order particle or fluid motion parameters, are required. A kth-order particle motion parameter P.sub.k (P for particle) is defined for k=1, 2, 3, . . . as the kth time derivative P.sub.k =d.sup.k x.sub.p /dt.sup.k of particle location or displacement in a specified direction x.sub.p. A kth-order fluid motion parameter F.sub.k (F for fluid) is similarly defined as F.sub.k =d.sup.k x.sub.f /dt.sup.k with x.sub.f a fluid-element location or displacement in a specified direction. Subscript k=1 denotes particle or fluid velocity P.sub.1 =dx.sub.p /dt or F.sub.1 =dx.sub.f /dt and subscript k=2 denotes particle or fluid acceleration P.sub.2 =d.sup.2 x.sub.p /dt.sup.2 or F.sub.2 =d.sup.2 x.sub.f /dt.sup.2, etc
Particle and fluid motions and an external force acting on a particle are related by Newton's second law of motion. By this law, particle and fluid properties and motions in any specified direction are related by a force balance equation or "particle equation of motion" by which the inertia, fluid friction, and net external forces acting on a particle are summed to zero, giving EQU m.sub.p .times.P.sub.2 =f.times.(F.sub.1 -P.sub.1)+G.sub.o [ 1]
or an equivalent equation, from which a family of equations is derived by taking k-1 successive time-derivatives to obtain, for constant m.sub.p and f, EQU m.sub.p .times.P.sub.k+1 =f.times.(F.sub.k -P.sub.k)+G.sub.k-1[ 2a]
or an equivalent equation, where [1] is given by [2a] when k=1 and thus k=1,2,3, or . . . , m.sub.p is the particle mass, f is the particle-fluid friction coefficient, and G.sub.o is an external force in the specified direction acting on the particle and G.sub.k-1 =d.sup.(k-1) G.sub.o /dt.sup.(k-1) is the (k-1)st time derivative of G.sub.o. Examples of G.sub.o are a gravitational force G.sub.o =m.sub.p .times.g, with g the component of the acceleration of gravity in the specified direction, and an electrostatic force G.sub.o =-q.times..epsilon., with q the electrostatic charge of the particle and .epsilon. the local electrostatic-field-strength component in the specified direction.
In the interest of simplicity in writing [2a], m.sub.p and f were regarded as constant in time. These assumptions are usually valid but are not essential and can be relaxed to give a more general form EQU d.sup.k [m.sub.p .times.P.sub.2 ]/dt.sup.k =d.sup.k =d.sup.k [f.times.(F.sub.1 -P.sub.1)+G.sub.o ]/dt.sup.k [ 2b]
or an equivalent equation. Both [2a] and [2b] are denoted [2] herein, with the simpler form [2a] generally used with the understanding that the simpler form represents both forms of [2].
Also in the interest of simplicity, the temporal nature of a signal is herein characterized only by time differences between signals or signal features, i.e., by signal time domain data. However, it is understood that a second type of data, i.e., signal frequency domain data, is equivalent in function and information content to the signal time-domain data, and that data of either type can be transformed to the other by known methods, some of which are described in the books entitled Mathematics of Physics and Modern Engineering by I. S. Sokolnikoff and R. M. Redheffer, 1958, and The Fourier Transform and Its Applications by R. N. Bracewell, second edition, 1986, both published by McGraw-Hill, New York. While the illustrations and claims herein are expressed in terms of signal time-domain data, it is understood that an equivalent description in terms of signal frequency-domain data is implicitly included throughout the specification and claims with the time-domain-data type because of the equivalence of the two types.
Quantities P.sub.1, P.sub.2, . . . , m.sub.p, f, and G.sub.o or various combinations of these quantities are categorized herein as particle, fluid, or particle-fluid properties, and some such as f or f/m.sub.p fit into two categories. One set of properties may be used to obtain, or are equivalent to, another, as illustrated in the following two examples.
(1) For slow, steady, vertical sedimentation of a particle (P.sub.2 =0) in a motionless fluid (F.sub.1 =0), [1] gives P.sub.1 =G.sub.o /f. Measured values of P.sub.1 and P.sub.2 with [1] thus provide a particle-fluid property G.sub.o /f. If the properties of the fluid (or particle) are known, G.sub.o /f provides a fluid-independent particle property (or vice versa). By use of G.sub.o =(m.sub.p -m.sub.f).times.g with m.sub.p =.pi./6.rho..sub.p D.sup.3 and m.sub.f =.pi./6.rho..sub.f D.sup.3 the mass of the particle of mass density .rho..sub.p and the mass of the fluid of known mass density .rho..sub.f displaced by the particle and 9.81 m/sec.sup.2 and by use of Stoke's law f=3.pi..eta..alpha.D, with .eta. a known fluid viscosity, .alpha. a particle shape correction factor, and D an equal-volume-sphere diameter for the particle, the result 18.eta.P.sub.1 /g=(.rho..sub.p -.rho..sub.f)D.sup.2 /.alpha. can be regarded as a particle, fluid, or particle-fluid property, i.e., independent of a fluid property when .eta. and .rho..sub.f are known, independent of a particle property when all such are specified, or dependent on both particle and fluid properties. In the case when .rho..sub.p &gt;&gt;.rho..sub.f, e.g., when the suspending fluid is a gas or rarefied gas, .rho..sub.p -.rho..sub.f is essentially equal to .rho..sub.p and a particle (or fluid) property called the Stoke's diameter D.sub.s =.sqroot.[.rho..sub.p D.sup.2 /.alpha.]=.sqroot.[18.eta.P.sub.1 /g] is obtained.
(2) When both m.sub.p .times.P.sub.2 /f and G.sub.o /f are negligible, a condition generally assumed in current LDV and LDA methods, [1] requires F.sub.1 =P.sub.1 and fluid velocity is accurately determined directly from a measured particle property, i.e., the particle velocity.
For simple motions, as in these two examples, simple and explicit relationships between measured and inferred quantities are obtained. For non-simple motions, such as when m.sub.p .times.P.sub.2 /f or G.sub.o /f is not negligible or is unknown as taught, for example, by Dahneke in U.S. Pat. No. 3,854,321, an elaborate, complex, case-specific calibration procedure must be used to relate a measured and an inferred quantity, a procedure requiring extensive calculations and additional information or assumptions.
Because of the assumptions m.sub.p .times.P.sub.2 /f and G.sub.o /f both &lt;&lt;F.sub.1 =P.sub.1, the accuracy of an inferred F.sub.1 value provided by current TOFV, TOFA, LDV, and LDA methods is generally undetermined but known to be poor in some cases. Similarly, the accuracy of a particle or fluid property inferred using at least one inferred F.sub.k value with particle equation of motion [1] or [2] is generally undetermined but known to be poor in some cases. Consequently, current methods are only reliable in a restricted range of application having ill-defined boundaries, with a complex and application-specific calibration procedure required for non-simple motions.
Error or uncertainty in a measured particle or fluid velocity obtained using current methods is often due, fundamentally, to failure of the assumption F.sub.1 =P.sub.1. Among practitioners, uncertainty in particle or fluid velocity is denoted "velocity broadening" and is attributed to specific causes including (1) velocity broadening due to time or space averaging, (2) velocity broadening due to mixed-property particles having mixed m.sub.p .times.P.sub.2 /f values and, therefore, mixed velocity difference (F.sub.1 -P.sub.1) values, and (3) velocity broadening due to poor signal quality (i.e., due to noisy signals).
In attempts to extend the usable range of current methods, small, low-inertia particles which more faithfully follow rapidly changing fluid motion have been used with the result that particles sufficiently small to significantly reduce velocity broadening due to cause (2) scatter much less light and provide poor signal quality, thus increasing velocity broadening due to cause (3). Increasing particle concentration does not help because more particles passing randomly through the sensing volume results in reduced variation in signal and reduced signal quality. Good signal quality, especially important in resolving rapidly changing signals encountered in high-rate-of-change particle and fluid motions like those in turbulent and rarefied gas or expanding-gas-jet flows, are not provided by current methods. These methods require small particle size to minimize velocity broadening by cause (2) and large particle size to minimize velocity broadening by cause (3), requirements that are mutually exclusive.
Because of limitations of current methods in determining motions or properties of a particle or fluid, improved methods are desired. These improved methods may be used to more accurately characterize the motion of one or more particles by providing at least two particle motion parameters, such as particle velocity P.sub.1 and acceleration P.sub.2. When used with equation of motion [1] or [2] to eliminate the assumption F.sub.1 =P.sub.1, improved accuracy may be obtained in an inferred fluid motion, an inferred property of a particle or fluid, or an inferred external force acting on a particle. When used multiple times with equation [1] or [2], multiple particle and fluid properties, such as particle charge q, friction coefficient f, and mass m.sub.p, can all be explicitly determined.
It is an object of this invention to provide an improved method and apparatus for characterizing motion of a particle by determining at least two particle motion parameters of the particle.
It is another object of this invention to provide an improved method and apparatus for determining motion of a fluid or of a fluid volume element by accurately determining at least one fluid motion parameter.
It is a further object of this invention to provide improved methods and apparatus for determining a particle property and to determine through multiple measurements of a particle the values of multiple properties of the particle, such as charge q, friction coefficient f, and mass m.sub.p.
It is an object of this invention to provide improved methods for determining one or more properties of a fluid.
It is another object of this invention to provide improved methods for measuring an external force acting on a particle in at least one specified direction or a time derivative of such a force.
It is an additional object of this invention to provide improved methods for characterizing a particle in simple and non-simple motions using simple calibration methods.